In the setting of the compressible Navier-Stokes equations, the corresponding wave \( (\bar{v}, \bar{u}, \bar{\theta }) \)  to the contact discontinuity \( (v^c, u^c, \theta ^c) \)  behaves as a diffusion wave due to the dissipation effect, which called a viscous contact wave (see [1]}, [2]}).
Now we construct the viscous contact wave \( (\bar{v}, \bar{u}, \bar{\theta }) \)  as follows.
Since the pressure for the profile \( (\bar{v}, \bar{u}, \bar{\theta }) \)  is hoped to be almost constant (see [3]}), we take
\(\bar{p}: = \frac{R \bar{\theta }}{\bar{v}} = p_+,\) 
